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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system


Authors: Mohammed Lemou, Florian Méhats and Pierre Raphaël
Journal: J. Amer. Math. Soc. 21 (2008), 1019-1063
MSC (2000): Primary 82C70, 35Q55, 35Q75, 85A05, 74H35
Published electronically: November 29, 2007
MathSciNet review: 2425179
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Abstract: The three dimensional gravitational Vlasov-Poisson system $ \partial_tf+v\cdot\nabla_x f-E_f\cdot\nabla_vf=0$, where $ E_f(x)=\nabla_x \phi_f(x)$, $ \Delta_x\phi_f=\rho_f(x)$, $ \rho_f(x)=\int_{\mathbb{R}^N} f(x,v)dv$, describes the mechanical state of a stellar system subject to its own gravity. Smooth initial data yield unique global in time solutions from a celebrated result by Pfaffelmoser. There exists a hierarchy of physical models which aim at taking into account further relativistic effects. The simplest one is the three dimensional relativistic gravitational Vlasov-Poisson system $ \partial_tf+\frac{v}{\sqrt{1+\vert v\vert^2}}\cdot\nabla_x f-E_f\cdot\nabla_vf=0$ which we study here. A striking feature as observed by Glassey and Schaeffer is that this system now admits finite blow up solutions. Nevertheless, the existence argument is purely obstructive and provides no insight into the description of the singularity formation. We exhibit in this paper a family of finite time blow up self-similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations. Our analysis applies to the four dimensional gravitational Vlasov-Poisson system as well.


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Additional Information

Mohammed Lemou
Affiliation: CNRS and Université Paul Sabatier, MIP, 118, Route de Narbonne, 31062 Toulouse, France

Florian Méhats
Affiliation: IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Pierre Raphaël
Affiliation: CNRS and Université Paris-Sud, Orsay, France

DOI: http://dx.doi.org/10.1090/S0894-0347-07-00579-6
PII: S 0894-0347(07)00579-6
Keywords: Vlasov-Poisson equations, relativistic gravitational kinetic equations, blow-up dynamics, virial law, modulation theory.
Received by editor(s): November 8, 2006
Published electronically: November 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society